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Trumpet Reincarnations: Proofs from the BookIntertranslatability Results for Abstract Argumentation Semantics
Christof Spanring
May 11, 2015
Christof Spanring Trumpet Reincarnations: Proofs from the Book
Building a Zoo
Which animals can share a pen?e1: s � b, snakes eat birds, e2: b � i , birds eat insects,e3: b � s, birds eat snakes, e4: i � i , insects eat insects,e5: f � b, felines eat birds, e6: f � f , felines eat felines.
s
b if
e1e3e2
e4
e5
e6
Groups with biggest impactsem(F ) = {{s}} stg(F ) = {{b}}
snakes defend themselves birds have biggest impact
Christof Spanring Trumpet Reincarnations: Proofs from the Book
Abstract Argumentation
Argumentation Frameworks (AF) [Dung, 1995]An AF F = (A,R) is composed of a set A of arguments and aset R ⊆ A× A of directed conflicts.Extensions E ⊆ A are specified by conditions such asconflict-freeness and maximality, a semantics σ(F ) is a specificcollection of extensions.
Language in UseFor a, b ∈ A, (a, b) ∈ R we say that a attacks b and writea � b.For E ⊆ A, a ∈ A we have E � a (resp. a � E ) if there issome e ∈ E such that e � a (resp. a � e).For any set E ⊆ A we call E+ = E ∪ {a ∈ A | E � a} therange of E .
Christof Spanring Trumpet Reincarnations: Proofs from the Book
Argumentation Semantics
Extension-Based SemanticsFor any given AF F = (A,R) and some set E ⊆ A we call E
conflict-free, iff there is no conflict between the arguments in E ;admissible, iff E is conflict-free and defends itself against allattacks from the outside;a stage extension, iff E is conflict-free and maximal withrespect to range;a semi-stable extension, iff E is admissible and maximal withrespect to range.
For any given AF we call a collection of some specific extensions asemantics, e.g. for semi-stable semantics we write sem(F ) for thecollection of all semi-stable extensions (which is a set of sets).
Christof Spanring Trumpet Reincarnations: Proofs from the Book
Intertranslatability
DefinitionA Translation from some semantics σ to some semantics σ′ is afunction Tr mapping AFs to AFs, we call Tr
exact: σ(F ) = σ(Tr (F )),in words: the original extensions and the new extensions areexactly the same;faithful: E ∈ σ(F ) iff ∃E ′, E ⊆ E ′, E ′ ∈ σ′(Tr (F )) and|σ(F )| = |σ′(Tr (F ))|,in words: we allow new arguments to expand the newextensions.
Christof Spanring Trumpet Reincarnations: Proofs from the Book
Sample Translations
s
exact Tr : stg ⇒ sem
b if
s
faithful Tr : sem ⇒ stg
bf i
h
stg(F ) = {{b}} sem(F ) = {{s}}sem(Tr 1(F )) = {{b}} stg(Tr 2(F )) = {{s, h}}
If birds can eat feline then If humans eat every animalstage becomes also semi-stable but snakes then semi-stable
becomes also stage.
Christof Spanring Trumpet Reincarnations: Proofs from the Book
There is no Exact Translation for sem⇒ stg
E ∈ stg(F ) ⇐⇒ E ∈ cf (F )∧ 6 ∃B ∈ cf (F ) : E+ ( B+
E ∈ sem(F ) ⇐⇒ E ∈ adm(F )∧ 6 ∃B ∈ adm(F ) : E+ ( B+
Counterexample
a3
a2 a1
c3
c1c2
b1 b2
b3
Christof Spanring Trumpet Reincarnations: Proofs from the Book
There is no Exact Translation for sem⇒ stg
Counterexample, Semi-Stable Extensions
a3
a2 a1
c3
c1c2
b1 b2
b3
sem(F ) = { {a2, b2}, {a1, b2, c1}, {a2, b1, c1}}
Christof Spanring Trumpet Reincarnations: Proofs from the Book
There is no Exact Translation for sem⇒ stg
Counterexample, Semi-Stable Extensions
a3
a2 a1
c3
c1c2
b1 b2
b3
sem(F ) = { {a2, b2}, {a1, b2, c1}, {a2, b1, c1}}
Christof Spanring Trumpet Reincarnations: Proofs from the Book
There is no Exact Translation for sem⇒ stg
Counterexample, Semi-Stable Extensions
a3
a2 a1
c3
c1c2
b1 b2
b3
sem(F ) = { {a2, b2}, {a1, b2, c1}, {a2, b1, c1}}
Christof Spanring Trumpet Reincarnations: Proofs from the Book
There is no Exact Translation for sem⇒ stg
Counterexample, Semi-Stable Extensions
a3
a2 a1
c3
c1c2
b1 b2
b3
sem(F ) = { {a2, b2}, {a1, b2, c1}, {a2, b1, c1}}
Christof Spanring Trumpet Reincarnations: Proofs from the Book
There is no Exact Translation for sem⇒ stg
Counterexample, Semi-Stable Extensions
a3
a2 a1
c3
c1c2
b1 b2
b3
sem(F ) = { {a2, b2}, {a1, b2, c1}, {a2, b1, c1}}
Christof Spanring Trumpet Reincarnations: Proofs from the Book
There is no Exact Translation for sem⇒ stg
a3
a2 a1
c3
c1c2
b1 b2
b3sem(F ) = {{a2, b2},
{a1, b2, c1},{a2, b1, c1}}
ProofAssume there is an exact translation Tr : sem⇒ stg , thenstg(Tr (F )) = sem(F ).
Christof Spanring Trumpet Reincarnations: Proofs from the Book
There is no Exact Translation for sem⇒ stg
a3
a2 a1
c3
c1c2
b1 b2
b3sem(F ) = {{a2, b2},
{a1, b2, c1},{a2, b1, c1}}
ProofAssume there is an exact translation Tr : sem⇒ stg , thenstg(Tr (F )) = sem(F ).It follows that c1 is not in conflict with neither a2 nor b2 inTr (F ).
Christof Spanring Trumpet Reincarnations: Proofs from the Book
There is no Exact Translation for sem⇒ stg
a3
a2 a1
c3
c1c2
b1 b2
b3sem(F ) = {{a2, b2},
{a1, b2, c1},{a2, b1, c1}}
ProofAssume there is an exact translation Tr : sem⇒ stg , thenstg(Tr (F )) = sem(F ).It follows that c1 is not in conflict with neither a2 nor b2 inTr (F ).Now {a2, b2, c1} is conflict-free in Tr (F ).
Christof Spanring Trumpet Reincarnations: Proofs from the Book
There is no Exact Translation for sem⇒ stg
a3
a2 a1
c3
c1c2
b1 b2
b3sem(F ) = {{a2, b2},
{a1, b2, c1},{a2, b1, c1}}
ProofAssume there is an exact translation Tr : sem⇒ stg , thenstg(Tr (F )) = sem(F ).It follows that c1 is not in conflict with neither a2 nor b2 inTr (F ).Now {a2, b2, c1} is conflict-free in Tr (F ).But then {a2, b2}+Tr (F ) ( {a2, b2, c1}+Tr (F ), and thus {a2, b2}can not be a stage extension in Tr (F ), a contradiction.
Christof Spanring Trumpet Reincarnations: Proofs from the Book
There is no modular faithful translation for sem⇒ stg
DefinitionA translation Tr is called modular iff from F = F1 ∪ F2 it followsthat also Tr (F ) = Tr (F1) ∪ Tr (F2), in words if we can build thetranslated framework by translating arbitrary parts, which is usefulfor distributed computing.
ObservationWe observe that a translation is called modular iff it is fully definedby translating the following frameworks:
(∅, ∅) ({a} , ∅)
a
({a} , {(a, a)})
a
({a, b} , {(a, b)})
a b
Christof Spanring Trumpet Reincarnations: Proofs from the Book
There is no modular faithful translation for sem⇒ stg
Counterexample
O4
a1
a2
a3
a4
O3
a1
a2 a3
Christof Spanring Trumpet Reincarnations: Proofs from the Book
There is no modular faithful translation for sem⇒ stg
Counterexample
Tr (a1, a2)
Tr (a2, a3)
Tr (a3, a4)
Tr (a4, a1)
O4
a1
a2
a3
a4
Tr (a1, a2)
Tr (a2, a3)
Tr (a3, a1)
O3
a1
a2 a3
Christof Spanring Trumpet Reincarnations: Proofs from the Book
There is no modular faithful translation for sem⇒ stg
stg(Tr (O3)) = {E}stg(Tr (O4)) = {E1, E2}
E ∩ AO3 = ∅
E1 ∩ AO4 = {a1, a3}
E2 ∩ AO4 = {a2, a4}
Tr (a1, a2)
Tr (a2, a3)
Tr (a3, a4)
Tr (a4, a1)
O4
a1
a2
a3
a4
Tr (a1, a2)
Tr (a2, a3)
Tr (a3, a1)
O3
a1
a2 a3
Proofsketch
Christof Spanring Trumpet Reincarnations: Proofs from the Book
There is no modular faithful translation for sem⇒ stg
stg(Tr (O3)) = {E}stg(Tr (O4)) = {E1, E2}
E ∩ AO3 = ∅
E1 ∩ AO4 = {a1, a3}
E2 ∩ AO4 = {a2, a4}
Tr (a1, a2)
Tr (a2, a3)
Tr (a3, a4)
Tr (a4, a1)
O4
a1
a2
a3
a4
Tr (a1, a2)
Tr (a2, a3)
Tr (a3, a1)
O3
a1
a2 a3
ProofsketchWe observe that E ∩ Tr (({ai , aj} , {(ai , aj)})) must be strictlyisomorphic for all attacks (ai , aj) ∈ O3. Simply because thereis only one extension of Tr (O3).
Christof Spanring Trumpet Reincarnations: Proofs from the Book
There is no modular faithful translation for sem⇒ stg
stg(Tr (O3)) = {E}stg(Tr (O4)) = {E1, E2}
E ∩ AO3 = ∅
E1 ∩ AO4 = {a1, a3}
E2 ∩ AO4 = {a2, a4}
Tr (a1, a2)
Tr (a2, a3)
Tr (a3, a4)
Tr (a4, a1)
O4
a1
a2
a3
a4
Tr (a1, a2)
Tr (a2, a3)
Tr (a3, a1)
O3
a1
a2 a3
ProofsketchWe observe that E ∩ Tr (({ai , aj} , {(ai , aj)})) must be strictlyisomorphic for all attacks (ai , aj) ∈ O3. Simply because thereis only one extension of Tr (O3).But then we can move E in an isomorphic extending way toTr (O4) receiving an unwanted extension. Namely an extensionthat does not contain any of a1, a2, a3, a4.
Christof Spanring Trumpet Reincarnations: Proofs from the Book
Dung, P. M. (1995).On the acceptability of arguments and its fundamental role innonmonotonic reasoning, logic programming and n-person games.Artif. Intell., 77(2):321–358.
Dvořák, W. and Spanring, C. (2012).Comparing the expressiveness of argumentation semantics.In Verheij, B., Szeider, S., and Woltran, S., editors, Proceedings ofthe 4th Conference on Computational Models of Argument (COMMA2012), volume 245 of Frontiers in Artificial Intelligence andApplications, pages 261–272. IOS Press.
Dvořák, W. and Woltran, S. (2011).On the intertranslatability of argumentation semantics.J. Artif. Intell. Res. (JAIR), 41:445–475.
Spanring, C. (2013).Intertranslatability results for abstract argumentation semantics.Master’s thesis, University of Vienna.
Christof Spanring Trumpet Reincarnations: Proofs from the Book